On homotopization of the unitary 𝐾₁-functor

Kopeĭko, V.

doi:10.1090/S1061-0022-09-01071-1

On homotopization of the unitary $K_1$-functor
HTML articles powered by AMS MathViewer

by V. I. Kopeĭko
Translated by: N. B. Lebedinskaya

St. Petersburg Math. J. 20 (2009), 749-755

DOI: https://doi.org/10.1090/S1061-0022-09-01071-1

Published electronically: July 21, 2009

PDF | Request permission

Abstract:

A unitary $K_1$-analog of the Karoubi–Villamayor functor is constructed, which solves the problem of homotopization of the unitary $K_1$-functor on the category of rings with involution.

References

Max Karoubi and Orlando Villamayor, Foncteurs $K^{n}$ en algèbre et en topologie, C. R. Acad. Sci. Paris Sér. A-B 269 (1969), A416–A419 (French). MR 251717
David L. Rector, $K$-theory of a space with coefficients in a (discrete) ring, Bull. Amer. Math. Soc. 77 (1971), 571–575. MR 292067, DOI 10.1090/S0002-9904-1971-12757-X
Jan R. Strooker, Karoubi-Villamayor $K$-theory is not homotopy Quillen theory, Proc. Amer. Math. Soc. 73 (1979), no. 2, 161–162. MR 516456, DOI 10.1090/S0002-9939-1979-0516456-9
Charles A. Weibel, Homotopy algebraic $K$-theory, Algebraic $K$-theory and algebraic number theory (Honolulu, HI, 1987) Contemp. Math., vol. 83, Amer. Math. Soc., Providence, RI, 1989, pp. 461–488. MR 991991, DOI 10.1090/conm/083/991991
Charles A. Weibel, Nil $K$-theory maps to cyclic homology, Trans. Amer. Math. Soc. 303 (1987), no. 2, 541–558. MR 902784, DOI 10.1090/S0002-9947-1987-0902784-0
Charles A. Weibel, KV-theory of categories, Trans. Amer. Math. Soc. 267 (1981), no. 2, 621–635. MR 626494, DOI 10.1090/S0002-9947-1981-0626494-7
Jens Hornbostel, $A^1$-representability of Hermitian $K$-theory and Witt groups, Topology 44 (2005), no. 3, 661–687. MR 2122220, DOI 10.1016/j.top.2004.10.004
Anthony Bak, $K$-theory of forms, Annals of Mathematics Studies, No. 98, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1981. MR 632404
Alexander J. Hahn and O. Timothy O’Meara, The classical groups and $K$-theory, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 291, Springer-Verlag, Berlin, 1989. With a foreword by J. Dieudonné. MR 1007302, DOI 10.1007/978-3-662-13152-7
Hyman Bass, Algebraic $K$-theory, W. A. Benjamin, Inc., New York-Amsterdam, 1968. MR 0249491
Max-Albert Knus, Quadratic and Hermitian forms over rings, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 294, Springer-Verlag, Berlin, 1991. With a foreword by I. Bertuccioni. MR 1096299, DOI 10.1007/978-3-642-75401-2
Ton Vorst, Polynomial extensions and excision for $K_{1}$, Math. Ann. 244 (1979), no. 3, 193–204. MR 553251, DOI 10.1007/BF01420342
Max Karoubi, Le théorème fondamental de la $K$-théorie hermitienne, Ann. of Math. (2) 112 (1980), no. 2, 259–282 (French). MR 592292, DOI 10.2307/1971147